In recent years, many scientific fields have experienced a data revolution. Fields such as human physical activity monitoring, neuroscience, cell biology, disease surveillance, and air quality recording have seen a proliferation of multivariate time-series data: datasets created by the repeated monitoring of a particular biological or physical system. Therefore, statistical models that can be used to effectively characterize interrelationships and interactions between these multiple time series, or data streams will prove useful across an increasingly wide range of scientific disciplines. In an infectious disease context, complex immunological interactions between multiple strains of disease govern the evolutionary and epidemiological dynamics of disease. For example, a common form of pathogen interaction is short-term immunity or cross-protection from subsequent infection. Another form of interaction is immune enhancement, where infection with one pathogen makes an individual more susceptible to infection or more likely to have severe clinical disease if infected. Understanding these interactions plays a vital role in clinical and public health decision-making. This knowledge informs the strategies for designing and testing new vaccines. It also advances our understanding of the mechanisms of disease transmission and progression at the individual and population levels. Specifically, this project aims to: (1) develop and fit a three-pathogen interaction model to influenza A, influenza B, and RSV surveillance data to estimate the duration and strength cross-protection between these three pathogens; (2) develop and fit a four-pathogen interaction model to dengue surveillance data to estimate the duration and strength of both cross-protection and enhancement between the four serotypes of dengue; and (3) develop and implement a statistical framework for comparing inferences from mechanistic and phenomenological models of multiple interacting time series. Using a model for cross-protection between pathogens, our work will provide the first explicit estimates of cross-protection between influenza a, influenza B, and RSV. Additionally, this work will provide the first explicit estimates of immune enhancement between the four serotypes of dengue. Therefore, this project will fill critical gaps in our knowledge about the epidemiology and immunology of two global disease systems. Fulfillment of the study aims will enhance our understanding of how to best draw statistical inference with mechanistic and phenomenological models in a wide array of scientific settings. Methods developed in the course of this work will be freely disseminated as R software packages on the Comprehensive R Archive Network, allowing other public health researchers and practitioners to learn from and build upon this work.